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The first two examples show that a subset of a locally compact space need not be locally compact, which contrasts with the open and closed subsets in the previous section. The last example contrasts with the Euclidean spaces in the previous section; to be more specific, a Hausdorff topological vector space is locally compact if and only if it ...
In mathematics, a locally compact group is a topological group G for which the underlying topology is locally compact and Hausdorff. Locally compact groups are important because many examples of groups that arise throughout mathematics are locally compact and such groups have a natural measure called the Haar measure .
The main examples of locally compact fields are the p-adic rationals and finite extensions /. Each of these are examples of local fields . Note the algebraic closure Q ¯ p {\displaystyle {\overline {\mathbb {Q} }}_{p}} and its completion C p {\displaystyle \mathbb {C} _{p}} are not locally compact fields [ 2 ] pg. 72 with their standard topology.
If G is a locally compact Hausdorff group, G carries an essentially unique left-invariant countably additive Borel measure μ called a Haar measure.Using the Haar measure, one can define a convolution operation on the space C c (G) of complex-valued continuous functions on G with compact support; C c (G) can then be given any of various norms and the completion will be a group algebra.
For every topological space Y, the projection is a closed mapping [11] (see proper map). Every open cover linearly ordered by subset inclusion contains X. [12] Bourbaki defines a compact space (quasi-compact space) as a topological space where each filter has a cluster point (i.e., 8. in the above). [13]
The product of a CG-1 space and a locally compact space is CG-1. [27] (Here, locally compact is in the sense of condition (3) in the corresponding article, namely each point has a local base of compact neighborhoods.) The product of a CG-2 space and a locally compact Hausdorff space is CG-2. [28] [29]
More precisely, let X be a topological space. Then the Alexandroff extension of X is a certain compact space X* together with an open embedding c : X → X* such that the complement of X in X* consists of a single point, typically denoted ∞. The map c is a Hausdorff compactification if and only if X is a locally compact, noncompact Hausdorff ...
A Hausdorff, Baire space that is also σ-compact, must be locally compact at at least one point. If G is a topological group and G is locally compact at one point, then G is locally compact everywhere. Therefore, the previous property tells us that if G is a σ-compact, Hausdorff topological group that is also a Baire space, then G is